13,961 research outputs found
Locating and Total Dominating Sets of Direct Products of Complete Graphs
A set S of vertices in a graph G = (V,E) is a metric-locating-total dominating set of G if every vertex of V is adjacent to a vertex in S and for every u ≠v in V there is a vertex x in S such that d(u,x) ≠d(v,x). The metric-location-total domination number \gamma^M_t(G) of G is the minimum cardinality of a metric-locating-total dominating set in G. For graphs G and H, the direct product G × H is the graph with vertex set V(G) × V(H) where two vertices (x,y) and (v,w) are adjacent if and only if xv in E(G) and yw in E(H). In this paper, we determine the lower bound of the metric-location-total domination number of the direct products of complete graphs. We also determine some exact values for some direct products of two complete graphs
On Covering Segments with Unit Intervals
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise
Bounding basic characteristics of spatial epidemics with a new percolation model
We introduce a new percolation model to describe and analyze the spread of an
epidemic on a general directed and locally finite graph. We assign a
two-dimensional random weight vector to each vertex of the graph in such a way
that the weights of different vertices are i.i.d., but the two entries of the
vector assigned to a vertex need not be independent. The probability for an
edge to be open depends on the weights of its end vertices, but conditionally
on the weights, the states of the edges are independent of each other. In an
epidemiological setting, the vertices of a graph represent the individuals in a
(social) network and the edges represent the connections in the network. The
weights assigned to an individual denote its (random) infectivity and
susceptibility, respectively. We show that one can bound the percolation
probability and the expected size of the cluster of vertices that can be
reached by an open path starting at a given vertex from above and below by the
corresponding quantities for respectively independent bond and site percolation
with certain densities; this generalizes a result of Kuulasmaa. Many models in
the literature are special cases of our general model.Comment: 15 page
- …